We study fixed-cardinality maximization of the inverse-matrix Solow--Polasky diversity, equivalently finite metric magnitude for the exponential kernel, on one-dimensional and ordered metric sets. The analysis starts from the known finite-line gap formula for the exponential kernel, which writes the excess inverse-matrix diversity as a sum of functions of consecutive gaps. Building on this formula, the main interval theorem proves that, for every $k\geq 2$, the unique maximizing $k$-point subset of $[0,1]$ is the equally spaced set. Thus the objective selects a uniform gap representation on the real line. A converse kernel proposition shows that, among normalized non-increasing distance kernels, requiring the corresponding adjacent-gap additive structure forces the exponential family. Further results transfer the interval theorem to ordered $\ell_1$ (L1, or Manhattan) curves by isometry: the maximizing sets are uniform in accumulated $\ell_1$ length. As a consequence, monotone biobjective Pareto fronts admit Solow--Polasky optimal finite approximations that are uniformly spaced in accumulated objective-space change, a natural representation when all parts of a continuous front should be covered. Examples, including a dense connected front and a finite disconnected ZDT3 front, illustrate how the continuous uniform-gap result appears on discrete candidate sets. Solow-Polasky diversity; diversity measures; finite metric magnitude; L1 distance; uniform spacing; Pareto-front approximation; multiobjective optimization; fixed-cardinality subset selection

翻译：我们研究了一维和有序度量集上逆矩阵Solow-Polasky多样性的固定基数最大化问题，等价于指数核下的有限度量振幅。分析始于指数核的已知有限线间隙公式，该公式将超额逆矩阵多样性表示为相邻间隙函数的和。基于该公式，主要区间定理证明：对于每个$k\geq 2$，$[0,1]$上唯一的最大化$k$点子集是等间距集。因此，目标函数在实线上选择均匀间隙表示。一个反向核命题表明，在标准化非递增距离核中，要求相应的相邻间隙可加结构会迫使核函数属于指数族。进一步的结论通过等距映射将区间定理推广到有序$\ell_1$（L1，或曼哈顿）曲线：最大化集在累积$\ell_1$长度上均匀分布。作为推论，单调双目标帕累托前沿允许Solow-Polasky最优有限逼近，这些逼近在累积目标空间变化上均匀间隔，当连续前沿的所有部分都应被覆盖时，这是一种自然表示。示例（包括密集连接的前沿和有限不连接的ZDT3前沿）说明了连续均匀间隙结果如何在离散候选集上出现。Solow-Polasky多样性；多样性度量；有限度量振幅；L1距离；均匀间距；帕累托前沿逼近；多目标优化；固定基数子集选择